In the previous section, we showed the full derivation of the closed-form solution to the marginal likelihood of model Mh. It turned out that there are four equivalent solutions to Equation.3. These solutions are derived by redefining which class map to the values 1 and 2 and by redefining which regions map toθ1andθ2.
Let us use the notations introduced in the previous section: a = N21+ α2, b = N22+ β2, c = N11+ α1 and d = N12+ β1. Also, let us define C as follows:
C =1 k· Γ(α1+ β1) Γ(α1) ·Γ(β1) · Γ(α2+ β2) Γ(α2) ·Γ(β2) (.9)
The marginal likelihood of Mh (Equation .3) can be obtained by solving any of the fol- lowing four equations:
C · a+b−1 X j=a Γ(a) ·Γ(b) Γ( j + 1) ·Γ(a + b − j)· Γ(c + j) ·Γ(a + b + d − j − 1) Γ(a + b + c + d − 1) (.10)
Which is the solution we derived in the previous section. C · d+c−1 X j=d Γ(c) ·Γ(d) Γ( j + 1) ·Γ(c + d − j)· Γ(b + j) ·Γ(c + d + a − j − 1) Γ(a + b + c + d − 1) (.11) C · Ã Γ(a) ·Γ(b) Γ(a + b) · Γ(c) ·Γ(d) Γ(c + d) − a+b−1 X j=b Γ(a) ·Γ(b) Γ( j + 1) ·Γ(a + b − j)· Γ(d + j) ·Γ(a + b + c − j − 1) Γ(a + b + c + d − 1) ! (.12) C · Ã Γ(a) ·Γ(b) Γ(a + b) · Γ(c) ·Γ(d) Γ(c + d) − c+d−1 X j=c Γ(c) ·Γ(d) Γ( j + 1) ·Γ(c + d − j)· Γ(a + j) ·Γ(c + d + b − j − 1) Γ(a + b + c + d − 1) ! (.13)
A.4 DERIVATION OF THE CLOSED-FORM SOLUTION FOR MODEL ML
The marginal likelihood of model Ml (P r(G|Ml)) is defined as follows:
=1 k Z 1 θ2=0 θ2N21· (1 − θ2)N22· beta(θ2;α2,β2) | {z } f1 Z θ2 θ1=0 θ1N11· (1 − θ1)N11· beta(θ1;α1,β1)dθ1 | {z } f2 dθ2 (.14) By solving the integral given by f2, we get:
f2= Γ(α1+ β1) Γ(α1) ·Γ(β1) Z θ2 θ1=0 θ1c−1· (1 − θ1)d−1dθ2 = Γ(α1+ β1) Γ(α1) ·Γ(β1)· Γ(c) ·Γ(d) Γ(c + d) · c+d−1 X j=c Γ(c + d) Γ( j + 1) ·Γ(c + d − j)· θ j 2· (1 − θ2)c+d−1− j
where, as before, c = N11+ α1and d = N12+ β1. By solving f1, we get: f1= Γ(α2+ β2) Γ(α2) ·Γ(β2) Z 1 θ2=0 θ2a−1· (1 − θ2)b−1 Now we can solve Equation.14:
P r(G|Ml) = C · c+d−1 X j=c Γ(c) ·Γ(d) Γ( j + 1) ·Γ(c + d − j)· Γ(a + j) ·Γ(c + d + b − 1 − j) Γ(a + b + c + d − 1) (.15) Where C is the constant we defined by Equation.9in the previous section.
Notice that Equation.15 (the solution to P r(G|Ml)) can be obtained from Equation .13 (one of the four solutions to P r(G|Mh)) as follows:
P r(G|Ml) = C ·Γ(a) ·Γ
(b)Γ(c) ·Γ(d)
Γ(a + b) · −Γ(c + d) − P r(G|Mh) (.16) It turned out that no matter which formula we used to solve P r(G|Mh), we can use Equation.16to obtain P r(G|Ml).
A.5 COMPUTATIONAL COMPLEXITY
Since we require that N11, N12, N21, N22,α1,β1,α2andβ2be natural numbers, the gamma function simply becomes a factorial function: Γ(x) = (x − 1)!. Since such numbers can be- come very large, it is convenient to use the logarithm of the gamma function and express Equations .2,.10, .11, .12, .13 and .16 in logarithmic form in order to preserve numerical precision. The logarithm of the integer gamma function can be pre-computed and efficiently stored in an array as follows:
lnGamma[1] = 0 For i = 2 to n
lnGamma[i] = lnGamma[i − 1] + ln(i − 1)
We then can use lnGamma in solving the above equations. However, Equations .10, .11,.12and.13include a sum, which makes the use of the logarithmic form more involved. To deal with this issue, we can define function lnAdd, which takes two arguments x and y that are in logarithmic form and returns ln(ex+ ey). It does so in a way that preserves a good deal of numerical precision that could be lost if ln(ex+ ey) were calculated in a direct
manner. This is done by using the following formula:
lnAdd(x, y) = x + ln(1 + e( y−x))
Now that we introduced functions lnGamma and lnAdd, it is straightforward to eval- uate Equations.2,.10,.11,.12,.13and.16in logarithmic form.
Let us now analyze the overall computational complexity for computing the Bayesian score for a specific rule (solving Equation .1). Doing so requires computing P r(Me|G), P r(Mh|G) and P r(Ml|G). P r(Me|G) can be computed in O(1) using Equation .2. P r(Mh|G) can be computed by applying Equation.10, Equation.11, Equation.12or Equation.13. The computational complexity of these equations are O(N22+ β2), O(N11+ α1), O(N21+ α2) and O(N12+ β1), respectively. Therefore, P r(Mh|G) can be computed in O(min(N11+ α1, N12+
β1, N21+ α2, N22+ β2). P r(Ml|G) can be computed from P r(Mh|G) in O(1) using Equation .16. By assuming that α1, β1, α2, β2 are bounded from above, the overall complexity for computing the Bayesian score is O(min(N11,N12,N21,N22).
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